A sequential procurement
model for a PPP project
pipeline
De Clerck D, Demeulemeester E.
KBI_1501
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A sequential procurement model for a PPP project pipeline
Dennis De Clerck1, Erik Demeulemeester KU Leuven, Faculty of Economics and Business
Department of Decision Sciences and Information Management Naamsestraat 69, B-3000 Leuven, Belgium
Tel: +32-16-32.67.58 / +32-16-32.69.72 Fax: +32-16-32.66.24
[email protected] [email protected]
Abstract
Public-private partnerships have seen the daylight in response to the adagio that project responsibilities and risks should be efficiently allocated between the public and the private sector. Nevertheless, the considerable bidding costs inhibit the competition in the market. A trustworthy project agenda could substantiate the PPP market’s attractiveness in the belief that a current success results in a knowledge and cost advantage in future tenders. This paper builds a sequential PPP procurement model and heuristically approximates the Markov perfect equilibrium in which contractors determine how much money they are willing to invest in the bid preparation and which mark-up is appropriate for each project in the pipeline. A pipeline of projects pushes down the mark-ups and the government procurement cost. Nonetheless, according to the experiments, it are only players with an initial experiential advantage who tend to make higher investment efforts so that additional governmental policies might be required to level the playing field.
Keywords: public-private partnerships, sequential procurement auction, Markov perfect equilibrium, best response heuristic
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1. Introduction
This paper offers a theoretical approach in analyzing public-private partnership (PPP) procurement for a sequence of projects. Since risk assessment and mitigation strategies continue to receive a great emphasis within the project management field, a PPP is a recent phenomenon that aims to attain an efficient risk allocation between a public sector entity and a private consortium. The latter party may, in addition to the construction, also engage in the design, maintenance and/or operation of the project. Due to the complex project nature and the high societal value that is at stake, financial performance requirements and high quality standards are important challenges one needs to face. Therefore, contractors ought to prepare a qualitative bid proposal to submit to the contracting government. The preparation of this proposal is costly (e.g., consultancy cost, working cost, design cost) and the risk of not being awarded the contract is empirically claimed to be a burden for contractors [1, 7, 25]. In order to ensure the competition, policy makers are endowed to seek for feasible ways to substantiate the PPP market’s attractiveness. This paper investigates whether a project pipeline succeeds in positively affecting the contractors’ bidding behavior. If the government agrees on a clear project agenda, contractors might be more eager to enter a particular market. Winning a contract might result in a competitive advantage in future tenders and, moreover, past losses can be recovered in future tenders.
This paper converts the PPP procurement setting into a sequential procurement auction format in which the government acts as the decision maker. For each project in the pipeline, the contractors or consortia that are invited for the tender determine their appropriate bid preparation effort willingness (i.e., the investment decision) and the mark-up percentage (i.e., the mark-up decision). Within this competitive framework, the bidders are heterogeneous in their experience levels, because winning a contract increases the experience in future tenders. The past experience a bidder has obtained together with the pre-tender investment efforts could lead to an increased ability to estimate the project cost on the one hand and to efficiency gains that result in a lower expected cost on the other hand. The study analytically characterizes the equilibrium of this stochastic game and employs an experimental setting to approximate the Markov perfect equilibrium for different project cost features and a varying number of projects in the pipeline. The method applies a best response heuristic for which an algorithmic approximation that is based on both an electromagnetism-like mechanism as well as a local search procedure
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efficiently detects the best response. This approach succeeds in looking at the entire, continuous action search space in order to identify candidate equilibria.
This research benefits practitioners and policy makers by offering a theoretical foundation for PPP procurement modeling. Next to its practical relevance, this work contributes to different stretches in the academic literature. First of all, competitive procurement models are scarce in the PPP literature while they could gain insights into a field in which empirical bidding data are often inadequately available. Moreover, theoretical models allow for an initial feasibility assessment of policy measures. Secondly, the PPP framework offers research opportunities within the OR community. The complex bidding framework is related to the traditional procurement auction field. In this vein, the paper’s model can be considered as an auction format with asymmetries among the bidders with respect to the project cost distribution and the option to invest in information acquisition. The pipeline concept adds a synergetic, sequential feature in the sense that past successes lead to a knowledge and cost advantage in future tenders. While the model’s complexities hinder a fully analytical approach, the algorithmic game theory assists in tackling the inclusiveness of the set-up.
The remainder of the paper is structured as follows. Section 2 explores the related operations research literature. Section 3 delves into the analytical foundation and the Markov equilibrium concept and subsequently discusses the experimental implementation and the equilibrium approximation algorithm. The outcomes of the scenario analysis are covered in Section 4 and are taken forward into a concluding discussion in Section 5.
2. Literature
The study of the sequential PPP procurement format is closely linked to the sequential auction theory where multiple units are auctioned in a strictly sequential fashion and where the winner is considered to be the highest bidder. The views on the expected price trends vary. Weber [48] proved that in the case of an uncertain common value, bids follow a Martingale, while the majority of theoretical studies argue that bid prices are declining. Examples include Von der Fehr [46] who studies auctions with participation cost and Kannan [20] in a setting with complete bid revelation. Branco [5] supports the price decline in the case of complementarities between identical objects, which is nuanced by Sørensen [41] in the case of a large number of stochastically equivalent objects or when probabilities to draw a large value are high. Engelbrecht-Wiggans [12] attributes this to the trade-off between a reduced competition effect
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and a second effect that is related to the number of chances to win. In addition, Menezes and Monteiro [27] relate the price trend to the type of synergies of objects. Hence, the sequential auction literature can be classified according to this object-related dimension. The objects may be complements and create synergies which has been extensively studied by Branco [5], Menezes and Monteiro [27] and De Silva et al. [10]. Alternatively, the objects can be homogeneous in nature in the sense that they are perfect substitutes [22, 53]. Finally, the objects may have the same stochastic feature and are said to be stochastically equivalent [12, 37-38]. Our model assumes that the projects have a constant stochastic nature, but contractors’ experience in past projects has an influence on the cost probability distribution of future projects. Apart from the type of the tendered objects, modeling approaches also differ in their consideration of the capacity constraints. Whereas Katzman [22] and Katehakis and Puranam [21] for instance do not include a capacity constraint, the contributions of Engelbrecht-Wiggans [12], Milgrom and Weber [29], Elmaghraby [11] and Reiβ and Schöndube [37] assume a single-unit demand. Other studies include monetary constraints [36] or restrict the availability of man-hours [43]. The PPP model does not account for capacity constraints and the results point towards rising mark-ups in later stages of the pipeline. Next to the mark-ups, contractors in the PPP model also make a decision on the pre-tender research efforts, but to the best of our knowledge, there have been no previous studies on the impact of a sequential mechanism on the willingness for information acquisition. The work of for instance Persico [35], Bergemann and Välimäki [3] and Shi [40] considers a single-shot game in which the bidders can gain information on the value of the auctioned object, but they do not extrapolate to a multi-unit environment.
More particularly, our paper is closely related to the contribution of Takano et al. [43] who study the competitive bidding strategy in a sequential setting with inaccurate cost estimates. Building upon earlier research from Naert and Weverberg [31] and King and Mercer [24], the authors explicitly account for the fact that bids are usually correlated with the estimated cost that is subject to inaccuracies. Takano et al. [43] apply a scenario-based approach for the cost estimates and a capacity constraint. Our work differs from and extends this study in several aspects. First of all, in addition to the mark-up choice, a bidder makes an investment decision that reduces the uncertainty and the expected cost. Secondly, Takano et al. [43] simulate costs and bids for the competitors, while we look at action equilibria by simultaneously optimizing
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each bidders’ pay-off. Thirdly, our model does not include capacity constraints or a value at risk constraint and deals with a constant number of heterogeneous contractors. Furthermore, it builds on the single-shot PPP procurement model of De Clerck and Demeulemeester [9] in three ways. Firstly, this paper extends the scope towards a multi-project environment. Besides, the current study considers a continuous strategy space and last but not least, the contractors’ pay-offs are not derived by a multi-agent simulation approach, but are exactly computed for presumed distributions.
The methodological approach is based on a dynamic programming model for which Markov equilibria are approximated based on a best response heuristic. The Markov perfect equilibrium concept, drawn from the study of Maskin and Tirole [26], is a solution concept that solves the stochastic game with a finite number of stages and where the pay-off and probabilistic transitions depend on the current state and the chosen actions [39]. The solution concept is not uncommon within the auction literature. In a sequential auction setting with randomly arriving bidders, Said [38] derives Markov equilibria to discuss bid shading (i.e., placing a bid below the estimated value) as a consequence of an option value of participation in future auctions. Within a procurement setting, Katehakis and Puranam [21] study the cost minimization objective of a buyer who wants to procure a fixed number of products and Yildirim [50] considers the optimal mechanism for piecewise procurement of large-scale projects. While these studies and our paper do not account for capacity constraints, Jofre-Bonet and Pesendorfer [18] include capacity limitations when defining the states of the game and they use empirical highway procurement bidding data to develop an estimation method for a repeated auction. Whereas most models allow for a different number of bidders in each stage, the PPP model assumes repeated competition among a constant set of bidders (like in [14] and [29]) and includes the current levels of the bidders’ past experience and the number of projects remaining in the state variable. Hence, the PPP model allows for heterogeneity among the bidders. The theoretical model of Jofre-Bonet and Pesendorfer [19] and the empirical studies by De Silva et al. [10] and Wolfram [49] account for endogenously appearing asymmetries due to synergies, but in the PPP setting bidders may ex ante differ in the cost probability distributions which is also the case in the work of Reiβ and Schöndube [37]. Moreover, in contrast to models that study infinite timeframes (e.g., [16, 33, 38, 52]), the PPP pipeline has a finite nature which is a logical consequence of the magnitude of the projects and the limited budget horizon of governments.
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The experimental set-up relies on a heuristic approach to derive the equilibrium. Algorithmic game theory attempts to deal with the complexity of real-life models [32]. Algorithms that use best response reasoning have been successfully implemented for instance for empirical games, routing games or dynamic oligopoly models and often limit the search space and the number of computationally expensive pay-off calculations (e.g., [13, 42, 47]). Nevertheless, a best response heuristic does not always converge and, to the best of our knowledge, formal proofs of convergence are limited to super-modular games with unique Nash equilibria [28] and congestion games [30]. The experimental results in this paper focus on the scenarios for which convergence has occurred.
Next to its contribution to the operations research field, this paper adds value to the PPP project management literature. The number of papers on PPPs has been proliferating and an overview of topics can be found in Ke et al. [23] and Tang et al. [44]. Past experiences have led to concise lists of success factors and key performance indicators (e.g., [17, 51]). Nevertheless, the majority of the studies are limited to attaining single-project success without looking at the broader PPP picture. However, empirical results also underline long-term and country-specific factors like the country’s government reputation, legal framework and economic stability [2, 8, 51] or the importance of PPP units in promoting PPPs [45]. In this vein, a consultancy report from KPMG [25] argues for efforts to levy barriers to competition and procurement inefficiencies. The report suggests that the pipeline concept could be beneficial, but this has not been covered within the academic PPP literature. Additionally, modeling the expensive bidding process has not received substantial consideration. Within this context, Zitron [54] developed a model to study the bid/no bid decision in function of the risk of bidding and the project risk itself. Especially for PPPs, the quality aspect of the bid proposals requires considerable attention. Without specifically referring to PPPs, Perng et al. [34] discuss a bidding game that experimentally assesses the economically most advantageous bidding setting. In a single-project setting, Ho [15] included government reimbursements to incentivize bidders to invest in more qualitative bids into a game-theoretical PPP bidding model. De Clerck and Demeulemeester [9] relate to Ho’s work, but include project contingencies and heterogeneous bidders into their bidding model. The authors claim that a bid cost reimbursement could be an effective tool to increase competition. Apart from the aforementioned procurement studies, the majority of the
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PPP studies consider a one-to-one relationship between a public entity and a single contractor to study pricing mechanisms and risk allocation negotiations [9].
3. Methodology
3.1Competitive bidding procedure
This paper extends the work of De Clerck and Demeulemeester [9] in which a single-project environment has been analyzed. The analysis of competitive bidding for PPP projects resulted in a complex procurement auction environment. In a stage 𝑧, contractors that are invited for the tender will first determine how much effort they are willing to put into the bid preparation. This investment may result in a reduction of the cost uncertainty as well as in a cost advantage. Moreover, the heterogeneity among contractors leads to more advantageous cost probability distribution functions for more experienced players, which means a smaller variance and a lower average expected cost. After the investment decision, each contractor 𝑝 estimates the project cost which is subject to estimation errors. Next, the mark-up is determined and applied to the estimated cost, resulting in the bid for project 𝑧. The lowest bidding contractor is granted the project. Afterwards, the tendering procedure for stage 𝑧 + 1 is initiated. The sequential model adds an additional feature: winning a project in the sequence results in additional experience for all subsequent projects.
3.2Sequential bidding model
Given is a commonly known project sequence 𝒵 ≔ {1,2, … , 𝑍}. We want to identify the strategy equilibrium of the subgame 𝑒 = (𝑒1, 𝑒2, … , 𝑒𝑃) that defines the initial experience setting of the players of the game. This setting is a stochastic game, as has been introduced by Shapley [39]. A stochastic game is a finite or infinite dynamic game that is played by one or more players with probabilistic transitions between a finite number of states. In this setting, the players are assumed to be long-lived and to have unlimited capacity to perform all the projects of the pipeline. In each stage of the sequential game, the contractors want to optimize their expected pay-off which consists of the instantaneous pay-off of the current stage and an expected continuation value of the pay-offs in future stages. We are looking at Markov strategies and identify a Markov perfect equilibrium (MPE) as presented by Maskin and Tirole [26]. This is justified as the current play is only influenced by the expected pay-offs of future projects on the one hand and the state variable on the other hand. For a given project or stage 𝑧 ∈ 𝒵 from the sequence, we determine the current state 𝜃𝑧∈ Θ𝑧 as the combination of the current experience
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levels and the number of remaining projects in the sequence 𝑍 − 𝑧 or 𝜃𝑧 = (𝑒, 𝑍 − 𝑧). The history ℎ𝑧 at stage 𝑧 is summarized into this state variable. Consequently, we assume that past investment and mark-up decisions solely impact the current behavior by having won or lost the tender. For a given state, a set of actions 𝒜𝑝𝑧 = 𝒜𝑝𝑧(𝜃𝑧) is available for each player. We assume that the set of available actions is the same in every state of the game, so that we can refer to the set of actions as 𝒜𝑝. An action 𝑎𝑝𝑧 in a state 𝜃𝑧 is composed of two elements: the amount of pre-tender investment 𝑖(𝑎𝑝𝑧|𝜃𝑧) a player 𝑝 is willing to adopt and which is expressed as a percentage of an initial cost base equal to 1 and the mark-up 𝑚(𝑎𝑝𝑧|𝜃𝑧) defined as a percentage that is
applied to the estimated project cost. The vector 𝑎𝑧= (𝑎1𝑧, … , 𝑎𝑃𝑧) represents the action profile or the combination of actions for all 𝑃 players. If we define the set of players as 𝒫 and the set of action profiles as 𝒜 =×𝑝∈𝒫𝒜𝑝, then we determine the transition probabilities 𝒬 from 𝒜 × Θz to Θz+1 so that 𝒬(𝜃𝑧+1|𝑎𝑧, 𝜃𝑧) represents the probability of arriving in state 𝜃𝑧+1 from the current state 𝜃𝑧 with an action profile 𝑎𝑧. For each stage 𝑧, the combination of a state 𝜃𝑧 and an action profile 𝑎𝑧 leads to an expected pay-off vector 𝜋𝑧(𝑎𝑧|𝜃𝑧) of dimension ℝ𝑃. Each player-specific element 𝜋𝑝𝑧(𝑎𝑧|𝜃𝑧) of this vector is the sum of an instantaneous pay-off 𝜌𝑝𝑧(𝑎𝑧|𝜃𝑧) and the player’s expected value 𝒱𝑝𝑧+1(𝑎𝑧|𝜃𝑧) for the future stages {𝑧 + 1, … , 𝑍} discounted with a factor 𝛿 (0 < 𝛿 ≤ 1). Consequently, the player-specific expected pay-off function that player 𝑝 wants to optimize at stage 𝑧 equals:
𝜋𝑝𝑧(𝑎𝑧|𝜃𝑧) = 𝜌𝑝𝑧(𝑎𝑧|𝜃𝑧) + 𝛿𝒱𝑝𝑧+1(𝑎𝑧|𝜃𝑧)
with
𝒱𝑝𝑧+1(𝑎𝑧|𝜃𝑧) = ∑ 𝒬(𝜃𝑧+1|𝑎𝑧, 𝜃𝑧)𝜋
𝑝𝑧+1(𝑎𝑧+1|𝜃𝑧+1) 𝜃𝑧+1∈Θ𝑧+1
and for each 𝑧 ∈ 𝒵 and 𝜃𝑧 ∈ Θ𝑧. This dynamic recursion is solved backwards, starting at 𝑧 = 𝑍 for all 𝜃𝑍 and defining 𝒱𝑝𝑍+1(𝑎𝑍|𝜃𝑍) = 0 for all 𝑝. Define 𝑎∗𝑍 as the resulting equilibrium action profile, so that 𝜋𝑝𝑍(𝑎∗𝑍|𝜃𝑍) is maximized. With this information, and assuming that players adopt equilibrium behavior for all subsequent games, the equilibrium for stage 𝑧 = 𝑍 − 1 can now be calculated. As soon as 𝑧 = 0, a strategy equilibrium for the sequential game has been obtained. Fig. 1 represents an example of an optimal strategy pattern and clarifies the notation.
[FIGURE 1 TO BE INSERTED HERE] 3.3Model assumptions
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The model assumptions of De Clerck and Demeulemeester [9] are equivalent in the sequential environment. First of all, we assume that the state of the game, which consists of the experience levels and the remaining projects in the pipeline, is common knowledge to all players. Moreover, the general risk structure of the projects is the same for all the projects in the pipeline. That means that, without considering experience and pre-tender investment, the ex ante cost probability distribution and its parameters do not change. The experience and the current investment level may change the shape of the cost probability distribution. The model only incurs project-specific investment efforts, so that investments only directly affect the current stage and do not contribute to the knowledge base of future stages. The experience as well as the investment have an impact on the expected value and the variance of the cost probability distribution. A user-defined experience scale [0,10] is defined with 𝑒𝑢 the number of intervals on the scale. Equivalently, 𝑒𝑢 refers to the number of projects won beyond which no extra
experience can be adopted. Consequently, the conversion of experience level 𝑒𝑝 of player 𝑝 on
the experience scale [0,10] is obtained by: 10 ∗𝑒𝑒𝑝 𝑢 = 𝑒𝑝
𝑐. If 𝑒
𝑢 = 2, an inexperienced player has 𝑒𝑝 = 𝑒𝑝𝑐 = 0 and a medium experienced and a highly experienced player have levels 𝑒
𝑝 = 1
(𝑒𝑝𝑐 = 5) and 𝑒𝑝= 2 (𝑒𝑝𝑐 = 10) respectively. The impact of experience and investment is implemented with diminishing scale effects and moreover, Gaussian distributions are assumed. Consequently, the contractor-specific project cost probability distribution 𝑐𝑝𝑧(𝑎𝑝𝑧|𝜃𝑧) is of the form 𝑁 (1 + 𝑔𝑝(𝑎𝑝𝑧|𝜃𝑧), 𝜎𝑝2(𝑎𝑝𝑧|𝜃𝑧)). We assume that 𝑔𝑝(. ) represents the fraction of the cost
that is accountable for the lack of experience or investment. The model assumes
𝑔𝑝(𝑎𝑝𝑧, 𝑒𝑝(𝜃𝑧)) = 𝛽𝑖𝑒−𝜇𝑖(100𝑖(𝑎𝑝
𝑧|𝜃𝑧))
+ 𝛽𝑒𝑒−𝜇𝑒𝑒𝑝
𝑐(𝜃𝑧)
which means that the expected value of the contractor-specific cost distribution at a particular state of the game is dependent on the amount of investment and the experience level. The parameters 𝛽𝑖 and 𝛽𝑒 reflect the maximum cost impact for players who do not invest and players without experience, respectively. 𝜇𝑖 and 𝜇𝑒 represent the associated growth parameters. Moreover, the variance of the distribution is dependent on the experience level and the investment. The total variance of the cost distribution is composed of an uncontrollable part that is equivalent for all players, a part attributed to the (lack of) experience and a fraction related to the (lack of) investment. Implementation-wise, we assume 𝜎𝑝2(𝑎𝑝𝑧, 𝑒𝑝(𝜃𝑧)) = 𝜎2+ (𝛾𝑖𝑒−𝜆𝑖(100𝑖(𝑎𝑝𝑧|𝜃𝑧)))
2
+ (𝛾𝑒𝑒−𝜆𝑒𝑒𝑝𝑐(𝜃𝑧))2 with 𝛾
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variance or risk that is caused by a lack of investment and 𝛾𝑒 related to the knowledge impact of experience and associated growth parameters 𝜆𝑖 and 𝜆𝑒. Assume, for instance that 𝑒𝑢 = 2,
𝜎 = 0.05, 𝛾𝑖 = 𝛾𝑒 = 𝛽𝑖 = 𝛽𝑒= 0.1, 𝜆𝑒 = 𝜇𝑒 = 𝜆𝑖 = 𝜇𝑖 = 0.25, then a player with no
experience and a 1% investment choice has a cost probability distributed as
𝑁(1.17788,0.01856) and for a player with experience level 1 (𝑒𝑝𝑐 = 5) and a 2% investment
choice, we get 𝑁(1.08930,0.00699). When a contractor 𝑝 estimates the project cost 𝐸̃𝑝𝑧, which
is generated from the cost probability distribution 𝑐𝑝𝑧(. ), he applies a mark-up to the expected cost to arrive at the bid (1 + 𝑚(𝑎𝑝𝑧|𝜃𝑧)) 𝐸̃𝑝𝑧. Consequently, the bid probability distribution given
a player’s action 𝑎𝑝𝑧 and a state 𝜃𝑧 equals 𝑏𝑝𝑧(𝑎𝑝𝑧|𝜃𝑧) and is characterized by the normal
distribution 𝑁 ((1 + 𝑚(𝑎𝑝𝑧|𝜃𝑧)) (1 + 𝑔𝑝(𝑎𝑝𝑧|𝜃𝑧)) , (1 + 𝑚(𝑎𝑝𝑧|𝜃𝑧))
2 𝜎𝑝2(𝑎
𝑝𝑧|𝜃𝑧)) with the
associated cumulative bid probability distribution 𝐵𝑝𝑧. It is assumed that the lowest bidding contractor is granted the project, so the probability of winning contract 𝑧 with 𝑃 players is
𝑞𝑝𝑧(𝑎𝑧|𝜃𝑧) = ∫−∞+∞𝑏𝑝𝑧(𝑥𝑝)[∏𝑘∈𝒫\{𝑝}(1 − 𝐵𝑘𝑧(𝑥𝑘))]𝑑𝑥𝑝. These probabilities also refer to the
transition probabilities 𝒬(𝜃𝑧+1|𝑎𝑧, 𝜃𝑧) to move from a state 𝜃𝑧 to 𝜃𝑧+1. The contractor that wins the project receives the proposed bid and pays the actual cost (1 + 𝑔𝑝(. )) of the project and the monetary investment effort. Additionally, this player obtains an updated experience level in all future stages of the game, i.e., until 𝑒𝑝 = 𝑒𝑢. The pay-off of the other contractors is determined by the lost investment. Nevertheless, the government might reimburse losing bidders for the investment efforts with a fraction 𝑑 (Section 4.5). Consequently, the instantaneous expected pay-off for a player 𝑝 in an action profile 𝑎𝑧 and state 𝜃𝑧 is given by:
𝜌𝑝𝑧(𝑎𝑧|𝜃
𝑧) = 𝑞𝑝𝑧(𝑎𝑧|𝜃𝑧) (𝐸[𝐵̃|𝑝 has won] − (1 + 𝑔𝑝𝑧 𝑝(𝑎𝑝𝑧|𝑒𝑝(𝜃𝑧))) − 𝑖(𝑎𝑝𝑧|𝜃𝑧)) − (1 − 𝑞𝑝𝑧(𝑎𝑧|𝜃𝑧)) (1 − 𝑑)𝑖(𝑎
𝑝𝑧|𝜃𝑧). 𝐸[𝐵̃|𝑝 has won]𝑝𝑧 refers to the ex post expected proposal
given that the player has won the tender and is found as the conditional expectation:
𝐸[𝐵̃|𝐵𝑝𝑧 𝑝𝑧 ̃ < 𝐵̃, ∀𝑘 ≠ 𝑝] = ∫𝑘𝑧 𝑥𝑝 𝑞𝑝𝑧𝑏𝑝 𝑧(𝑥 𝑝) ∏𝑘∈𝒫\{𝑝}(1 − 𝐵𝑘𝑧(𝑥𝑝)) +∞
−∞ 𝑑𝑥𝑝. In order to arrive at the
total expected pay-off for this state, the discounted player’s values for the future stages need to be added to the expected instantaneous pay-off.
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In each state, we define the equilibrium action profile that determines the equilibrium investment percentage and mark-up percentage for the players. Finding this equilibrium is a hard problem and a heuristic approach is developed. The equilibrium of the dynamic programming problem is first determined for the states that are related to the final project in the pipeline. Based on a backward induction reasoning, the equilibria for each stage 𝑧 are determined, assuming equilibrium play in the stages {𝑧 + 1, … , 𝑍}. So for a state variable 𝜃𝑧 = (𝑒1, 𝑒2, … , 𝑒𝑃, 𝑍 − 𝑧) that is composed of the experience levels of all players and the number of remaining projects in the pipeline, we want to derive an equilibrium action profile 𝑎∗ = (𝑖1, 𝑚1, … , 𝑖𝑃, 𝑚𝑃), which is a vector with dimension ℝ2∗𝑃. If no confusion is possible, subscripts and superscripts related to the state variable are omitted so as to avoid notational complexity. In order to determine the equilibrium, a straightforward best response heuristic is applied. In each iteration of the heuristic, the algorithm approaches the best response for each player sequentially, given the actions of the opponents. Once none of the bidders can improve their response given the opponents’ actions, there is evidence that one might have arrived in an equilibrium. Academic literature only offers theoretical proofs for a limited number of circumstances for which the best response algorithm converges [6, 47]. Also in the context of the PPP procurement setting, we did not succeed in theoretically guaranteeing the convergence of the heuristic, while the computational results indicated convergence for the majority of the investigated cases. Due to the complex structure of the pay-off function and the infinite set of possible actions, the identification of the best response is challenging and time consuming. Therefore, the heuristic uses a combination of an electromagnetic algorithm and a local search procedure. It is important to note that there is no guarantee that a pure strategy equilibrium exists in the case of an infinite number of actions. The algorithm looks into unique equilibria, but of course multiple equilibria might exist and the convergence towards a particular equilibrium is path-dependent. Therefore, the algorithm is executed for a predefined number of starting points. This repetitive structure accounts for the possibility of arriving in alternative equilibria. The pseudo-code for the algorithms can be found in Appendix 1.
3.4.1 Best response determination
Optimizing the pay-off function for a given set of actions for the opponents (indicated as 𝑎−𝑝)
is a computationally intensive task because of the structure of the non-linear pay-off response function (Fig. 2). There are two decision variables for each player 𝑝 that, together, represent the
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action 𝑎𝑝: the investment percentage 𝑖𝑝 and the mark-up percentage 𝑚𝑝. The best response
optimizes argmax𝑎𝑝𝜋𝑝𝑧(𝑎𝑝|𝜃𝑧, 𝑎−𝑝 ) = 𝜌𝑝𝑧(𝑎𝑝|𝜃𝑧, 𝑎−𝑝) + 𝛿𝒱𝑝𝑧+1(𝑎𝑝|𝜃𝑧, 𝑎−𝑝). In order to efficiently determine the global optimum, this paper uses an electromagnetism-like mechanism for global optimization as has been proposed by Birbil and Fang [4] and whose notation and procedure is utilized here. The attraction-repulsion mechanism of the heuristic succeeds in efficiently browsing through the entire search space and overcomes the danger of arriving in a local minimum, which could be the case when adopting the steepest ascent heuristic. The derivation of the best response is a two-step process. Given the action profile of the opponents
𝑎−𝑝, a set 𝒯 of 𝑇 solutions is initialized. A solution 𝑡 ∈ 𝒯 is represented as 𝑥𝑡 = (𝑥1𝑡, 𝑥2𝑡), with 𝑥1𝑡 ∈ [𝑙
1, 𝑢1] the coordinate that refers to the investment level and 𝑥2𝑡 ∈ [𝑙2, 𝑢2] the coordinate
that refers to the mark-up level. 𝑙1 and 𝑙2 represent the lower bounds and 𝑢1 and 𝑢2 the upper bounds. For each initial point, the expected pay-off for player 𝑝, for whom the best response is derived, equals 𝑓𝑝(𝑥𝑡) = 𝜋𝑝(𝑥𝑡|𝜃𝑧, 𝑎−𝑝). The best solution is stored as 𝑓𝑝(𝑥best).
[FIGURE 2 TO BE INSERTED HERE] 3.4.1.1Electromagnetism step
A charge is calculated for each point, based on the superposition principle of electromagnetism theory. The charge represents point 𝑖’s power of attraction or repulsion and is calculated by: 𝑞𝑡 = exp (−2 ∗ (𝑓𝑝(𝑥
best)−𝑓 𝑝(𝑥𝑡))
∑𝑇𝑘=1(𝑓𝑝(𝑥best)−𝑓𝑝(𝑥𝑘))
). In the next step, the total forces on each point are computed. For the pairwise force calculations, the point that has a higher expected pay-off, attracts the other point, while the point with a worse pay-off repels the other point. The forces and moves are calculated for all but the currently best found point, in order to keep the information of the current best point. Furthermore, let 𝐹𝑛𝑡𝑢 represent the force exerted by point 𝑢 on point 𝑡 for coordinate 𝑛, with 𝑛 = 1 for the investment direction and 𝑛 = 2 for the mark-up direction. Finally, 𝐹𝑛𝑡 refers to the total force exerted by the other points for coordinate 𝑛. As a result, we obtain 𝐹𝑡= {𝐹1𝑡, 𝐹2𝑡} = {∑𝑢∈𝒯\{𝑡}𝐹1𝑡𝑢, ∑𝑢∈𝒯\{𝑡}𝐹2𝑡𝑢} and the force is represented as:
𝐹𝑡 = ∑ (−1)𝑤
𝑡𝑢
(𝑥𝑢− 𝑥𝑡)𝑞𝑡𝑞𝑢 ‖𝑥𝑢 − 𝑥𝑡‖2 𝑢∈𝒯\{𝑡}
with 𝑤𝑡𝑢 = 0 if 𝑓𝑝(𝑥𝑡) < 𝑓𝑝(𝑥𝑢) and 𝑤𝑡𝑢 = 1 if 𝑓𝑝(𝑥𝑡) ≥ 𝑓𝑝(𝑥𝑢). In order to move the points,
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𝑈(0,1) so that there is a non-zero probability to move to the unvisited regions along the direction of 𝐹𝑡. The normalization of the force vector ensures that the new solution is located in the feasible region. In conclusion, all but the previously found best point are moved so that for each
𝑡 ∈ 𝒯\{best} and 𝑛 ∈ {1,2}, the respective coordinate becomes:
𝑥𝑛′𝑡 = {𝑥𝑛 𝑡 + 𝜂𝐹̿
𝑛𝑡(𝑢𝑛− 𝑥𝑛𝑡) if 𝐹̿𝑛𝑡> 0 𝑥𝑛𝑡 − 𝜂𝐹̿
𝑛𝑡(𝑥𝑛𝑡 − 𝑙𝑛) if 𝐹̿𝑛𝑡 ≤ 0
The procedure is iterated a predefined number of times with the newly found coordinates and subsequently, the local search algorithm is executed.
3.4.1.2Example
Consider a two-player situation for which we want to determine the best response for 𝑝 = 1 and given the action (1,20) for player 2 which refers to a 1% investment percentage and a 20% mark-up percentage. Table 1 reports the algorithm values for 𝑇 = 4. The pay-off of player 1 for each point 𝑡 is given by 𝑓1(𝑥𝑡), so that the best pay-off is obtained in point 1. The sum of all the differences between the points’ pay-off values and the best pay-off is 0.035589 and the charge for point 𝑡 = 2 is found as exp (−2 ∗0.046640−0.0404480.035589 ). The distance between point 2 and 3 is 4.4778 and the force that point 3 exerts on point 2 equals 𝐹123 = (1.7 − 3.5) ∗0.7484∗0.70614.4778 =
−0.2125 for the investment coordinate and 𝐹223 = (16.1 − 20.2) ∗0.7484∗0.70614.4778 = −0.4839 for the mark-up coordinate. The forces on point 2’s investment coordinate from the points 1 and 4 are -0.1022 and 0.0419 respectively, so that the total force 𝐹12 = −0.2728. After normalizing and generating 𝜂=0.4 as the random variable and defining (𝑙1, 𝑙2) = (0,0) and (𝑢1, 𝑢2) = (5,30),
the new investment coordinate for 𝑡 = 2 equals 3.5 + 0.4 ∗ (−0.5725) ∗ 3.5 = 2.6985. The electromagnetism procedure is iterated with the previously best found point (𝑡 = 1) and with the three newly found points.
[TABLE 1 TO BE INSERTED HERE] 3.4.1.3Local search step
The points that result from the electromagnetism step are the input for a local search procedure around each of these points. In each iteration, the investment and mark-up coordinate are modified by applying a random transformation around the original point using a step length vector 𝛼 and a randomly generated point 𝜅 from a uniform distribution that determines the amount and direction of the perturbation. If the new point results in a higher pay-off, the
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coordinates are updated and the local search procedure continues from the newly found point. In the end, the best known solution is updated.
3.4.1.4Example
Let us start from the initial point (1.48,21.37) while the action for the second player is
(1,20). This results in a pay-off of 0.049456. Assume that the step length for the investment is 0.3 and 2 for the mark-up. For each coordinate, a random variable is generated from 𝑈(0,1):
𝜅 = 0.09 for the investment coordinate and 𝜅 = 0.86 for the mark-up coordinate. In order to allow for positive and negative perturbations, we subtract 0.5 from the randomly generated number, so the new investment coordinate becomes 1.48 + (0.09 − 0.5) ∗ 0.3 = 1.36 and for the mark-up shift, the mark-up in the new point equals 21.37 + (0.86 − 0.5) ∗ 2 = 22.09. The point (1.36,22.09) leads to a pay-off of 0.0501 and consequently the local search continues from this outperforming point.
3.4.2 Equilibrium selection
For a starting point that is randomly selected from the entire action profile space, the best response heuristic is executed until the convergence criterion is satisfied. Convergence occurs as soon as no significant pay-off improvements have been registered in a pre-defined number of loops and as long as a minimum number of loops 𝐵𝑅𝑀𝐼𝑁𝑅𝐸𝑃 has been executed. Moreover, the algorithm stops when the number of loops reaches 𝐵𝑅𝑀𝐴𝑋𝑅𝐸𝑃. The convergence is path-dependent, so the algorithm is repeated for 𝑅 starting points. Afterwards, the 𝑅 resulting action profiles are clustered according to a distance criterion resulting in 𝐶 clusters. Points are sequentially assigned to the cluster for which the distance between the point and the average coordinates of the cluster is minimal and smaller than 𝑀𝐴𝑋𝐷𝐼𝑆𝑇. Alternatively, a new cluster is formed. Within each cluster, the coordinate-by-coordinate average of all points belonging to the cluster are calculated together with the pay-off profile. Clusters with a number of points smaller than 𝑀𝐼𝑁𝑆𝑈𝑃𝑃𝑂𝑅𝑇 are removed as these might have resulted from a local optimum. The second step in the equilibrium selection process compares the averages of the clusters and removes any clusters that are dominated by other clusters (i.e., a cluster that has higher pay-offs for all players). Subsequently, the clusters are ranked according to the sum of the average pay-offs of all players and the cluster with the highest total pay-off is moving to the refinement stage. The refinement stage executes the best response heuristic again to confirm whether the particular
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action profile is an equilibrium. If this is confirmed, the resulting reported equilibrium for the state is stored and is used to compute the equilibria in the stage 𝑧 − 1.
3.5Experimental setting
The computer experiments have been executed in Microsoft Visual Studio 2010 and Wolfram Mathematica 8.0 performs the pay-off calculations. As the risk profile of PPP projects deserves major interest, we are mainly interested in the effect of the level of uncertainty, which is reflected by the knowledge impact parameters 𝛾𝑖 and 𝛾𝑒. Moreover, it is necessary to account for the computational effort that the equilibrium determination requires. Therefore, three sets of scenarios have been developed. Each scenario is executed for every unique combination of experience levels. Sections 4.2 and 4.3 give insights into the general dynamics of the equilibrium and rely on the four scenarios that are determined by the parameter values of Table 2. We will refer to these as follows: Sc.1 (𝛾𝑖 = 𝛾𝑒 = 0.1), Sc. 2 (𝛾𝑖 = 0.1, 𝛾𝑒 = 0.05), Sc. 3 (𝛾𝑖 = 0.05, 𝛾𝑒 =
0.1) and Sc. 4 (𝛾𝑖 = 𝛾𝑒 = 0.05). Secondly, Section 4.4 covers a case (Sc. 5) with extreme risks
(𝛾𝑖 = 0.2, 𝛾𝑒 = 0.1). Thirdly, the impact of introducing government compensation in Sc. 1 and Sc. 5 is assessed in Section 4.5. Finally, Section 4.6 highlights some noteworthy robustness aspects from a more extensive sensitivity analysis of the cost impact related parameters in a setting with only two experience intervals (𝑒𝑢 = 2). Besides, after trading off computation times and the level of accuracy and assessing the speed of convergence, the algorithm variables are tuned as shown in Table A1.1 of Appendix 1.
[TABLE 2 TO BE INSERTED HERE] 4. Experimental results
4.1Algorithm performance
For each state of the game, the algorithm looks for the state-specific action equilibrium. As the algorithm is based on a best response heuristic, it is important to assess the convergence performance. A scale with 𝑒𝑢 = 5 intervals leads to six experience levels, which results in 21 experience vectors (0,0), (0,1), (0,2), (0,3), (0,4), (0,5), (1,1), (1,2), …, (5,5) in a two-player setting. Equivalently, a three-player setting with five intervals results in 56 experience vectors. Convergence occurs when the algorithm stops before the maximum number of replications has been reached. For Sc. 1-4, the convergence criterion is met up to a three-project pipeline (𝑍 = 3) in the three-player setting. In the two-player setting, convergence issues appeared solely in a three-project pipeline for subgames (0,4) and (0,5) in which experienced players are randomizing
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within a small area of the action profile space of Sc. 1 between actions without investment on the one hand and with a small investment (0.5%) and a slightly higher (1%) mark-up.
In the convergent cases, given the parameters in Table A1.1, the search procedure converges already after less than 𝐵𝑅𝑀𝐼𝑁𝑅𝐸𝑃 loops towards a specific search region in the action profile matrix. Changing the players’ sequence of best response selection does not modify this tendency. In the first loops of the best response algorithm, the electromagnetism is mainly attributable for the selection of the best response, while in later iterations the local search procedure will be mainly responsible for the determination of the optimizing step. In the vast majority of the cases, the algorithm reports a single cluster. Only in the two- and three-project context of Sc. 1 and Sc. 2, two equilibria are apparent for the mature markets (4,4,5) and (4,5,5). These equilibria are characterized by a zero investment by one of the equally experienced players, while the other does not invest.
4.2Impact of the pipeline on the first project procurement [TABLE 3 TO BE INSERTED HERE]
Table 3 reports the statistical results of the scenario-by-scenario analysis for the strategic actions in the tender for the first project in a single-, a two- and a three-project environment with two and three players. Within the stochastic game notation, the table compares the actions of the states for a constant experience vector 𝑒, but for a variable number of remaining projects in the pipeline. So, if a state is represented as 𝜃𝑧 = (𝑒1, 𝑒2, 𝑍 − 𝑧), this section compares the actions for
𝜃1 = (. , . ,0) with 𝜃1 = (. , . ,1) and 𝜃1 = (. , . ,2). A parametric paired t-test has been used to
study the paired observations. In order to guarantee the results’ robustness, the output of the non-parametric Wilcoxon signed rank test is also reported. The statistical tests confirm the initial expectation that mark-ups are lowered when extra projects are introduced both in the two-player as well as in the three-player case. The drop in mark-ups is the greatest for the inexperienced players, resulting in more competition within the heterogeneous market. An analysis of the investment dynamics tends to point towards decreasing investment percentages when more projects are in the pipeline. Conditioning on the experience levels for the comparison of a three- versus a single-project case with three players indicates an insignificant average absolute drop of 0.001% for inexperienced players value=0.106), towards a significant average 0.02% drop (p-value=0.004) for players with 𝑒1 = 1 and 𝑒1 = 2 and a drop of 0.045% for players with maximum experience (p-value=0.0003). Looking at the two-player environment in which the
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investments are generally higher, the results show a significant decrease of 0.24% and 0.05% for players with experience levels 0 and 1 respectively (p-values 0.001 and 0.025), but do report a significant increase for players from experience level 3 onwards.
Eventually, the decreasing mark-ups also lead to lower bidder’s pay-offs for the first project and a lower project procurement cost for the government. According to the aggregated scenario outcomes, the mark-up and consequently also the pay-off drop is largest for the inexperienced players and in the subgames where mainly inexperienced players are involved. On average, a decrease of 13.53% in the pay-off has been reported in the two-player case and 19.35% in the three-player case when moving from a single-project to a three-project pipeline. Additionally, inexperienced players are sometimes willing to suffer a loss in the first project in order to win the project and obtain a greater experience level for the next project. In general, the percentage-wise drop in pay-offs has a negative trend in the experience level. The reduced mark-ups have positive repercussions on the government expenditures. The government cost for the first project in a three-project pipeline is 1.12% lower in two-player subgames and 0.92% lower in three-player subgames than in the case without a pipeline. In immature markets, savings of 5.14% and 4.13% might be realized in two- and three-player settings respectively.
Furthermore, an ANOVA study of the investment percentage in function of the model parameters and the state-defining values with first-order interaction effects could not support significance at the 5% level of any of the terms related to the number of projects in the pipeline in the three-player case. In the two-player model, both the interaction term of the experience level 𝑒1 and the number of projects in the pipeline as well as the interaction term of the competitor’s experience level 𝑒2 and the pipeline length are highly significant (p-values equal 4.6*10-7 and 9.8*10-5 respectively). The ANOVA models for the mark-ups, on the other hand, reveal the main effect and the interaction effects of the pipeline variable. We refer to Appendix 2 for the full ANOVA results. The ANOVA tests also underline the interaction with the players’ experience vector that defines the state variable. Tables 4 and 5 show the action dynamics in the bidding behavior for the first project for a player with experience level 𝑒1 with respect to a variable number of projects in the pipeline. Only the scenarios that involve players with experience level zero up to two are reported, but analogous results are apparent for the remaining scenarios. Moving towards a multi-project environment considerably reduces the mark-ups for the inexperienced players. According to these tables, only players with a competitive advantage
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over all their competitors (e.g., the subgames (𝑒1, 𝑒2, 𝑒3)=(1,0,0), (2,0,0) or (2,0,1) and
(𝑒1, 𝑒2)=(1,0), (2,0) or (2,1)) have a tendency to invest more in a multi-project environment than in a single-project setting, which means that players who were already in a beneficial position opt to strengthen their advantage even more. The cases in which a player has an experience level that is equal to or lower than at least one of his opponents point towards decreasing investment percentages. Also for maximally experienced players, we do not find significance for increasing investment levels, regardless of the experience level of the opponents.
[TABLES 4 AND 5 TO BE INSERTED HERE] 4.3Impact of a pipeline on the average expected bidding behavior
Section 4.2 solely looked at the bidding dynamics of the first project of the pipeline. A second approach to analyze the dynamics of the bidding behavior compares the average expected investment and mark-up percentages of the players over the entire pipeline. Given the optimal action profiles 𝑎∗𝑧, the average investment level over the entire pipeline is calculated by taking the sum of the investment levels of each state 𝜃𝑧 = (. , . , 𝑍 − 𝑧) for each 𝑧 weighted with the probability that this state occurs. Consider the three-project pipeline and initial experience vector
(𝑒1, 𝑒2) = (0,0) of Fig. 1. The expected average investment percentage for player 1 equals (0.6% + 0.43(1.3% + 0.50 ∗ 1.6% + 0.50 ∗ 1.3%) + 0.57(0.6% + 0.43 ∗ 1.3% + 0.57 ∗ 0.8%))/3 ≈ 0.9%. These averages are then compared to the situation in which a single project is tendered three times, which means that only the actions and transition probabilities of the state variables 𝜃𝑧 = (. , . ,1) are considered. The average expected pay-offs and government cost may be obtained in a similar fashion. Table 6 reports the statistical analysis of the differences and agrees with the previously stated findings. The former mark-up results are confirmed so that, from a procurement perspective, bidding becomes more aggressive in the case of a project pipeline than when projects are tendered without communicating the pipeline. The analysis of the average investment efforts in the experiment show that the pipeline concept does on average not incur extra investment efforts. Nonetheless, conditioning on the experience level reveals the experience-dependent response. The differences in the dynamics are also attributed to the competitive position of the player with respect to the opponents. In the model, pre-tender investment results in a reduction of the uncertainty, so that sometimes more accurate estimations point towards higher project costs. Consequently, bidders might prefer to be less informed and play with the mark-up, win a project and use the experience for future projects.
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[TABLE 6 TO BE INSERTED HERE] 4.4High-risk situation
A change of the 𝛾𝑖 parameter of Sc. 1 to a value of 0.2 refers to a case with high project-related risk (Sc. 5). The observation of the three-player equilibrium results has led to three findings. Firstly, not all subgames could guarantee convergence. In the subgames (0,0,4) and (0,0,5) with a single project for instance, the inexperienced players are looping over strategy profiles in which they both participate or one stays out. Consequently, we might expect that there is an equilibrium in mixed strategies. As the results of a multi-project pipeline rely on the outcome of the single-project case, this non-convergence effect acts as a bullwhip to earlier states of the stochastic game. Secondly, among all scenarios, there are only two cases for which the equilibrium allows participation for all players for the single-project case: subgame (0,0,0) with a 2.1% investment and a 25% mark-up for all participants and subgame (1,1,1) with a 2.4% investment and a 21% mark-up. In all other subgames, the reported equilibrium always suggests that one player should stay out of this engagement (i.e., 0% investment and 50% mark-up). Thirdly, adding extra projects has the same consequences as described earlier, but players who applied a no-participation action in the single-project case, will still prefer to stay out of the market. As a result, the model argues that it might be unsustainable to invite three contractors for the bid preparation stage in a high-risk project environment. Therefore, the equilibria of a two-player setting are of very high interest. In this vein, the previous results seem to be robust: everyone participates, mark-ups decrease in the length of the pipeline and players with a competitive advantage over their opponent tend to invest more, but the investment gaps are small. Hence, levelled strategic behavior is expected if uncertainty is highly project-dependent.
4.5Government reimbursement
In general, one might say that the introduction of a pipeline leads to fiercer competition from a mark-up perspective. Nevertheless, the inexperienced bidders are still significantly less willing to invest in research, which is especially the case in a three-player setting. Therefore, a government reimbursement might help to level the playing field from an investment perspective. The government determines a compensation percentage 𝑑 that reflects the fraction of the investment efforts that is reimbursed to all losing bidders. Table 7 summarizes the main characteristics of the equilibrium results for Sc. 1 and high-risk Sc. 5 in a two-player setting and for Sc.1 in the three-player case. The figures use a similar approach as in Section 4.3, so that we
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look at the average bidding behavior, the average pay-off and the average government expenditure over the entire three-project pipeline.
[TABLE 7 TO BE INSERTED HERE]
If two players are prequalified, the investment willingness is relatively symmetric for both players, both in the low-risk as well as in the high-risk case. An introduction of a compensation raises the investment willingness. The average mark-up percentages tend to decrease to the level of 40% compensation in the low-risk case, while they increase from 40% compensation onwards in the high-risk setting. Of course, this reimbursement involves additional expenditures for the government, but these extra costs can be partly offset by the savings that result from the pipeline concept. In the low-risk case with a three-project pipeline, a compensation between 20% and 40% would lead to optimal results: an investment increase and a government expenditure decrease. Interestingly, the average government expenditure in the high-risk setting is lower than in the low-risk environment. Moreover, both players have a considerable investment willingness, making the introduction of a reimbursement policy obsolete, because it would only increase the government cost and inflate the contractors’ profits.
Three-player competition will generally lead to fiercer competition and a lower government cost. Nevertheless, Table 7 shows that without compensation, it rarely happens that all players of a particular subgame invest. The use of compensation overcomes this dynamic in the majority of the subgames, resulting in a higher average investment. In the subgames where inexperienced players are competing against two mature contractors though, the inexperienced player moves to an equilibrium action in which he does not participate. On average, a compensation of 60% together with a pipeline of three projects is cheaper than a situation without compensation and without a pipeline. So, in the case governments aim for three competing consortia, a combination of both policies leads to promising results. Section 4.4 highlights the mixed equilibrium behavior of the high-risk case, which prevents us from giving a full analysis of the compensation mechanism. Nevertheless, the experiments show that an 80% reimbursement levies the convergence issue and would ensure that all players of the subgame invest in 46 out of the 56 subgames. For the ten remaining vectors, the 80% reimbursement does not serve as a sufficient incentive to make the inexperienced player who is facing two players with an 𝑒𝑝 of at least 2 to invest more. Without a compensation, it is only the case in two out of 56 subgames that all participants invest, while one player does not participate in all the other subgames. Nevertheless,
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in this high-risk setting, it is very expensive to attribute these compensations. Consequently, this analysis suggests the policy to prequalify only two contractors in a high-risk setting, while a three-contractor environment with an investment reimbursement incurs optimal results in a low-risk setting. As the investment percentages reflect the willingness of the contractor to invest, it should evidently be the government’s priority to reduce the (non-value adding) investment requirements.
4.6Additional scenarios
In order to investigate the robustness of the results, a full factorial 24 design has been set up with two levels (i.e., 0.05 and 0.1) for the parameters 𝛾𝑖, 𝛾𝑒, 𝛽𝑖 and 𝛽𝑒. The scenarios have been
executed for each subgame (i.e., every possible combination of the experience levels) in a two- and three-player setting with 𝑍 = {1,2,3}. For this robustness study, only two experience intervals are considered, so that 𝑒𝑢 = 2 and a player’s experience level is 0, 1 or 2.
First of all, the results for the states with experience vectors (0,2) and (0,0,2) often did not converge (i.e., in 11 out of 16 scenarios and 5 out of 16 scenarios respectively). In these instances, the algorithm loops between different action profiles. The inexperienced players mix their choice between no investment and a high mark-up and a moderate investment with a low mark-up. The experienced player responds accordingly with a low or a high mark-up. It is especially when the parameters that are related to the experiential advantage (𝛾𝑒 and 𝛽𝑒) are high that these convergence issues occur. Nevertheless, the dynamics of the previous sections are confirmed by this sensitivity study. Mark-ups are decreasing in the number of projects, regardless of the number of players and their respective experience levels. Concerning the investments, the players who have a competitive advantage in the initial stage of the game increase the investment willingness when the project pipeline grows larger.
The limited number of experience intervals underlines the importance of winning a project early in the pipeline. Therefore, the percentage-wise impact of the pipeline on the mark-ups is greater than when 𝑒𝑢 = 5. According to the aggregated scenario outcomes, the pay-off drop for the first project of a three-project pipeline is largest for the inexperienced players and in subgames in which inexperienced players are involved. On average, a decrease of 73% in the pay-off has been reported in the two-player case and 109% in the three-player case when moving from a single-project to a three-project pipeline. Consequently, inexperienced players are willing to suffer a loss in the first project in order to win it and obtain a greater experience level for the
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next project. The pay-off drops for the medium experienced player equal 20% and 33% for the two- and three-player setting respectively and for the player with maximum experience, this amounts to an 8% and a 17% drop. As a result, the combined sensitivity results also confirm the reduction in the government expenditures. A three-project pipeline reduces the procurement cost of the first project with 1.7% in subgames with two players and with 2.2% in three-player subgames.
4.7Practitioners’ view
Along the course of the research project, a number of international contracting companies, Belgian and Australian public institutions and a set of advisory firms have been interviewed to provide feedback on the models and the results. They all believe in the positive impact of a project pipeline and its capability of increasing competition. For contractors, it is a means to spread budgets and risk over multiple projects, although mature companies believe that the pipelines will limit their profits because of the stronger competitive forces. Another consequence of the theoretical findings is that governments might consider to split up large-scale infrastructure projects into a sequence of smaller projects. As a result, also small contractors, for whom the bid costs might become a financial burden, benefit from entering the market. If a government wants to ensure to keep the competition going and to avoid a situation in which contractors become too comfortable in their mature position, they might have to put effort in cementing the pipeline so that contractors have sufficient belief in the trustworthy pipeline. In addition, practitioners also underline the importance of reducing the bidding cost in general and of investing in the efficiency of the procurement process [25].
5. Conclusion
This paper studies the impact of a pipeline on the dynamic bidding behavior in a PPP setting. Governments do not always have the opportunity to foresee the budget for these risky and expensive projects in the long term. Nevertheless, public institutions in Canada and Australia for instance do succeed in maintaining and cementing a considerable project pipeline. The dynamic procurement format has been translated into a sequential procurement auction model in which contractors that are heterogeneous in their cost probability distributions will determine their sequential strategy over the course of a commonly known pipeline and a constant set of bidders. These bidders can, at each stage, modify their ex ante expected cost distributions by their investment decision. Being granted a project results in a knowledge and cost advantage for future
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projects. An experimental study has been performed to gain initial insight into the optimal strategy. A method to gradually derive the Markov equilibrium has been presented. Under the assumptions that outline the scope of the paper, the results support the hypothesis that mark-ups tend to decrease when more projects are included in the pipeline, regardless of the number of bidders. In this vein, the mark-up result is in line with the contemporary sequential auction theory literature. Consequently, bidders are willing to accept lower profits if this might lead to future benefits. Moreover, we have assumed that the continuation value of the final project has been set to zero, which actually even amplifies this finding. As a consequence, the government procurement cost tends to have a decreasing nature in the number of projects. Looking at the upfront investment willingness, there has been only limited support for an increasing trend for players who dispose of a competitive advantage at the start of the pipeline. In all other cases, the investment percentages tend to decrease. Therefore, an investment reimbursement might still be necessary to trigger the consortia’s enthusiasm to perform more upfront research. This levels the competition and reduces the risk of a contractor’s default, which would always come at the government’s expense due to the societal value of PPP projects. The extra cost of investment reimbursements might be partly offset by the reduced mark-ups that result from the pipeline. Nevertheless, in cases with considerable project-specific risks, it might be better to only prequalify two contractors to engage in the expensive tendering process. Contractors, public institutions and advisors that have provided comments on these results all believe in the importance of a project pipeline to increase competition and to attract new or overseas consortia that are willing to invest more money and resources to enter a market with a long-term perspective.
Methodology-wise, this study has some limitations that require further consideration. Firstly, the findings are theoretical, are based on stringent assumptions and essentially need to be validated in a practical environment. As a PPP setting is highly competitive, bidding data are scarce to fully validate the model. Furthermore, a relaxation of the assumptions that pre-tender investment does not contribute to future knowledge and that contractors have unlimited capacity offers an opportunity for further research. Thirdly, for some scenarios, the heuristic did not converge in all states of the game. As earlier stages of the pipeline rely on the optimal bidding behavior of later stages, this limits the study of the entire scenario. Subsequently, the flexibility of the model trades off against the manageability of the analytical calculations, so that
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form expressions or convergence proofs are not possible. Nevertheless, this shortcoming together with the highly relevant and state-of-the-art bidding framework, leaves opportunities for research in related fields like mathematical auction theory and algorithmic game theory. Nonetheless, the flexibility of the model does make it easy to test new governmental policies and the impact on the bidding behavior. Last but not least, the magnitude and the financial burden of PPP projects often inhibit to bid on more than one or two projects. Further research could focus on the interdependency of capacity constraints and the resulting equilibria.
Acknowledgement
The authors would like to address their gratitude towards the Intercollegiate Center for Management Science who has provided the financial resources to fund this research project. We would like to mention the support and hospitality of the University of South Australia and the feedback of all practitioners who have delivered insightful comments to the work.
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